Fourth Edition, last update April 19, 2007
2
Lessons In Electric Circuits, Volume V – Reference

                    By Tony R. Kuphaldt

          Fourth Edition, las...
i

   c 2000-2009, Tony R. Kuphaldt
   This book is published under the terms and conditions of the Design Science License...
ii
Contents

1 USEFUL EQUATIONS AND CONVERSION FACTORS                                                                       ...
iv                                                                                                                        ...
CONTENTS                                                                                                                  ...
Chapter 1

USEFUL EQUATIONS AND
CONVERSION FACTORS

Contents
     1.1     DC circuit equations and laws . . . . . . . . . ...
2                         CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS

         1.11 Data . . . . . . . . . . . . ....
1.2. SERIES CIRCUIT RULES                                                                       3

1.2      Series circuit...
4                           CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS

1.4.2       Series and parallel inductance...
1.5. CAPACITOR SIZING EQUATION                                          5

   ε = ε0 K

      Where,
         ε0 = Permitt...
6                          CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS

1.6        Inductor sizing equation

     N...
1.7. TIME CONSTANT EQUATIONS                                                                7

  The inductance in henries...
8                         CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS

1.7.2    Calculating voltage or current at s...
1.8. AC CIRCUIT EQUATIONS                                                             9

1.8.2     Capacitive reactance

 ...
10                           CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS

1.8.6         Resonance

                ...
1.9. DECIBELS                                         11

1.9     Decibels
                                             AV...
12                       CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS

1.10       Metric prefixes and unit conversion...
1.10. METRIC PREFIXES AND UNIT CONVERSIONS                                                   13

  • Conversion factors fo...
14                                CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS

     Conversion equivalencies for pr...
1.10. METRIC PREFIXES AND UNIT CONVERSIONS                                                     15

For example, 2500 calor...
16                          CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS


          Original figure       175 gallo...
Chapter 2

COLOR CODES

Contents

        2.1   Resistor Color Codes        . . . . . . . . . . . . . . . . . . . . . . . ...
18                                                              CHAPTER 2. COLOR CODES


      Color     Digit Multiplier ...
2.1. RESISTOR COLOR CODES                                                                     19

2.1.1    Example #1



 ...
20                                                               CHAPTER 2. COLOR CODES

2.2     Wiring Color Codes
Wiring...
2.2. WIRING COLOR CODES                                                                    21


                     Table...
22                                                              CHAPTER 2. COLOR CODES

by Wiles. [2] He makes no recommen...
Chapter 3

CONDUCTOR AND INSULATOR
TABLES

Contents
      3.1   Copper wire gage table . . . . . . . . . . . . .         ....
24                           CHAPTER 3. CONDUCTOR AND INSULATOR TABLES

7 ---------- 0.1443 ------- 20,820   ------ 0.0163...
3.3. COEFFICIENTS OF SPECIFIC RESISTANCE                                             25

|           TW               RUH ...
26                                    CHAPTER 3. CONDUCTOR AND INSULATOR TABLES

3.4       Temperature coefficients of resi...
3.6. DIELECTRIC STRENGTHS FOR INSULATORS                                                  27

Critical temperatures, high ...
28   CHAPTER 3. CONDUCTOR AND INSULATOR TABLES
Chapter 4

ALGEBRA REFERENCE

Contents
     4.1     Basic identities . . . . . . . . . . .     .   .   .   .   .   .   .  ...
30                                                      CHAPTER 4. ALGEBRA REFERENCE

4.1         Basic identities
a+0=a  ...
4.4. RADICALS                                                                                   31

4.4           Radicals...
32                                                      CHAPTER 4. ALGEBRA REFERENCE




      e=           1
            ...
4.7. FACTORING EQUIVALENCIES                                                                 33

4.6.2     Properties of l...
34                                                     CHAPTER 4. ALGEBRA REFERENCE

4.8       The quadratic formula

For ...
4.10. FACTORIALS                                                                            35

4.9.2      Geometric seque...
36                                                      CHAPTER 4. ALGEBRA REFERENCE

      x + y = 24
     2x - y = -6
  ...
4.11. SOLVING SIMULTANEOUS EQUATIONS                                                        37

           x + y = 24



 ...
38                                                    CHAPTER 4. ALGEBRA REFERENCE

                x=6
                  ...
4.11. SOLVING SIMULTANEOUS EQUATIONS                                                          39

    3(y - z + 10) + y + ...
40                                                     CHAPTER 4. ALGEBRA REFERENCE

obtain a figure for z:
          y=4
 ...
4.11. SOLVING SIMULTANEOUS EQUATIONS                                                            41

side of the = sign), a...
42                                                    CHAPTER 4. ALGEBRA REFERENCE

we seek:
     -2(3x + y) = -2(13)

   ...
4.11. SOLVING SIMULTANEOUS EQUATIONS                                                         43

        x - y + z = 10
  ...
44                                                     CHAPTER 4. ALGEBRA REFERENCE

         x - y + z = 10

            ...
4.12. CONTRIBUTORS                                                                             45

allows us to solve for ...
46   CHAPTER 4. ALGEBRA REFERENCE
Chapter 5

TRIGONOMETRY REFERENCE

Contents
        5.1    Right triangle trigonometry . . . . . . .        .   .   .   . ...
48                                               CHAPTER 5. TRIGONOMETRY REFERENCE

5.1.1    Trigonometric identities

 si...
5.3. TRIGONOMETRIC EQUIVALENCIES                                                             49

5.2.2       The Law of Co...
50                                          CHAPTER 5. TRIGONOMETRY REFERENCE

   Harvey Lew (??? 2003): Corrected typogra...
Chapter 6

CALCULUS REFERENCE

Contents

     6.1   Rules for limits . . . . . . . . . . . . . . . . . . . . . . . . . . ....
52                                                      CHAPTER 6. CALCULUS REFERENCE

6.1         Rules for limits
lim [f...
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  1. 1. Fourth Edition, last update April 19, 2007
  2. 2. 2
  3. 3. Lessons In Electric Circuits, Volume V – Reference By Tony R. Kuphaldt Fourth Edition, last update April 19, 2007
  4. 4. i c 2000-2009, Tony R. Kuphaldt This book is published under the terms and conditions of the Design Science License. These terms and conditions allow for free copying, distribution, and/or modification of this document by the general public. The full Design Science License text is included in the last chapter. As an open and collaboratively developed text, this book is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the Design Science License for more details. Available in its entirety as part of the Open Book Project collection at: www.ibiblio.org/obp/electricCircuits PRINTING HISTORY • First Edition: Printed in June of 2000. Plain-ASCII illustrations for universal computer readability. • Second Edition: Printed in September of 2000. Illustrations reworked in standard graphic (eps and jpeg) format. Source files translated to Texinfo format for easy online and printed publication. • Third Edition: Equations and tables reworked as graphic images rather than plain-ASCII text. • Fourth Edition: Printed in XXX 2001. Source files translated to SubML format. SubML is a simple markup language designed to easily convert to other markups like LTEX, HTML, A or DocBook using nothing but search-and-replace substitutions.
  5. 5. ii
  6. 6. Contents 1 USEFUL EQUATIONS AND CONVERSION FACTORS 1 1.1 DC circuit equations and laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Series circuit rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Parallel circuit rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Series and parallel component equivalent values . . . . . . . . . . . . . . . . . . 3 1.5 Capacitor sizing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6 Inductor sizing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.7 Time constant equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.8 AC circuit equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.9 Decibels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.10 Metric prefixes and unit conversions . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.11 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.12 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 COLOR CODES 17 2.1 Resistor Color Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Wiring Color Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 CONDUCTOR AND INSULATOR TABLES 23 3.1 Copper wire gage table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Copper wire ampacity table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Coefficients of specific resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4 Temperature coefficients of resistance . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.5 Critical temperatures for superconductors . . . . . . . . . . . . . . . . . . . . . . 26 3.6 Dielectric strengths for insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.7 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4 ALGEBRA REFERENCE 29 4.1 Basic identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Arithmetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Properties of exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.4 Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.5 Important constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 iii
  7. 7. iv CONTENTS 4.6 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.7 Factoring equivalencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.8 The quadratic formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.9 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.10 Factorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.11 Solving simultaneous equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.12 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5 TRIGONOMETRY REFERENCE 47 5.1 Right triangle trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Non-right triangle trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.3 Trigonometric equivalencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.4 Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.5 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6 CALCULUS REFERENCE 51 6.1 Rules for limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.2 Derivative of a constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.3 Common derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.4 Derivatives of power functions of e . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.5 Trigonometric derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.6 Rules for derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.7 The antiderivative (Indefinite integral) . . . . . . . . . . . . . . . . . . . . . . . . 55 6.8 Common antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.9 Antiderivatives of power functions of e . . . . . . . . . . . . . . . . . . . . . . . . 56 6.10 Rules for antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.11 Definite integrals and the fundamental theorem of calculus . . . . . . . . . . . . 56 6.12 Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7 USING THE SPICE CIRCUIT SIMULATION PROGRAM 59 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 7.2 History of SPICE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.3 Fundamentals of SPICE programming . . . . . . . . . . . . . . . . . . . . . . . . 61 7.4 The command-line interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.5 Circuit components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.6 Analysis options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.7 Quirks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 7.8 Example circuits and netlists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 8 TROUBLESHOOTING – THEORY AND PRACTICE 113 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 8.2 Questions to ask before proceeding . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.3 General troubleshooting tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.4 Specific troubleshooting techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.5 Likely failures in proven systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.6 Likely failures in unproven systems . . . . . . . . . . . . . . . . . . . . . . . . . . 123
  8. 8. CONTENTS v 8.7 Potential pitfalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 8.8 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 9 CIRCUIT SCHEMATIC SYMBOLS 129 9.1 Wires and connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 9.2 Power sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 9.3 Resistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 9.4 Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 9.5 Inductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 9.6 Mutual inductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 9.7 Switches, hand actuated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 9.8 Switches, process actuated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 9.9 Switches, electrically actuated (relays) . . . . . . . . . . . . . . . . . . . . . . . . 136 9.10 Connectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 9.11 Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9.12 Transistors, bipolar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 9.13 Transistors, junction field-effect (JFET) . . . . . . . . . . . . . . . . . . . . . . . . 138 9.14 Transistors, insulated-gate field-effect (IGFET or MOSFET) . . . . . . . . . . . . 139 9.15 Transistors, hybrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 9.16 Thyristors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 9.17 Integrated circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 9.18 Electron tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 10 PERIODIC TABLE OF THE ELEMENTS 145 10.1 Table (landscape view) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 10.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 A-1 ABOUT THIS BOOK 147 A-2 CONTRIBUTOR LIST 151 A-3 DESIGN SCIENCE LICENSE 155 INDEX 158
  9. 9. Chapter 1 USEFUL EQUATIONS AND CONVERSION FACTORS Contents 1.1 DC circuit equations and laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Ohm’s and Joule’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Kirchhoff ’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Series circuit rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Parallel circuit rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Series and parallel component equivalent values . . . . . . . . . . . . . . 3 1.4.1 Series and parallel resistances . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4.2 Series and parallel inductances . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4.3 Series and Parallel Capacitances . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Capacitor sizing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6 Inductor sizing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.7 Time constant equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.7.1 Value of time constant in series RC and RL circuits . . . . . . . . . . . . 7 1.7.2 Calculating voltage or current at specified time . . . . . . . . . . . . . . . 8 1.7.3 Calculating time at specified voltage or current . . . . . . . . . . . . . . . 8 1.8 AC circuit equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.8.1 Inductive reactance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.8.2 Capacitive reactance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.8.3 Impedance in relation to R and X . . . . . . . . . . . . . . . . . . . . . . . 9 1.8.4 Ohm’s Law for AC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.8.5 Series and Parallel Impedances . . . . . . . . . . . . . . . . . . . . . . . . 9 1.8.6 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.8.7 AC power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.9 Decibels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.10 Metric prefixes and unit conversions . . . . . . . . . . . . . . . . . . . . . . 12 1
  10. 10. 2 CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS 1.11 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.12 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.1 DC circuit equations and laws 1.1.1 Ohm’s and Joule’s Laws Ohm’s Law E = IR I= E R= E R I Joule’s Law 2 P = IE P= E P = I2R R Where, E = Voltage in volts I = Current in amperes (amps) R = Resistance in ohms P = Power in watts NOTE: the symbol ”V” (”U” in Europe) is sometimes used to represent voltage instead of ”E”. In some cases, an author or circuit designer may choose to exclusively use ”V” for voltage, never using the symbol ”E.” Other times the two symbols are used interchangeably, or ”E” is used to represent voltage from a power source while ”V” is used to represent voltage across a load (voltage ”drop”). 1.1.2 Kirchhoff’s Laws ”The algebraic sum of all voltages in a loop must equal zero.” Kirchhoff’s Voltage Law (KVL) ”The algebraic sum of all currents entering and exiting a node must equal zero.” Kirchhoff’s Current Law (KCL)
  11. 11. 1.2. SERIES CIRCUIT RULES 3 1.2 Series circuit rules • Components in a series circuit share the same current. Itotal = I1 = I2 = . . . In • Total resistance in a series circuit is equal to the sum of the individual resistances, mak- ing it greater than any of the individual resistances. Rtotal = R1 + R2 + . . . Rn • Total voltage in a series circuit is equal to the sum of the individual voltage drops. Etotal = E1 + E2 + . . . En 1.3 Parallel circuit rules • Components in a parallel circuit share the same voltage. Etotal = E1 = E2 = . . . En • Total resistance in a parallel circuit is less than any of the individual resistances. Rtotal = 1 / (1/R1 + 1/R2 + . . . 1/Rn ) • Total current in a parallel circuit is equal to the sum of the individual branch currents. Itotal = I1 + I2 + . . . In 1.4 Series and parallel component equivalent values 1.4.1 Series and parallel resistances Resistances Rseries = R1 + R2 + . . . Rn 1 Rparallel = 1 1 1 R1 + R2 + . . . Rn
  12. 12. 4 CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS 1.4.2 Series and parallel inductances Inductances Lseries = L1 + L2 + . . . Ln 1 Lparallel = 1 1 1 L1 + L2 + . . . Ln Where, L = Inductance in henrys 1.4.3 Series and Parallel Capacitances Capacitances 1 Cseries = 1 1 1 C1 + C2 + . . . Cn Cparallel = C1 + C2 + . . . Cn Where, C = Capacitance in farads 1.5 Capacitor sizing equation εA C= d Where, C = Capacitance in Farads ε = Permittivity of dielectric (absolute, not relative) A = Area of plate overlap in square meters d = Distance between plates in meters
  13. 13. 1.5. CAPACITOR SIZING EQUATION 5 ε = ε0 K Where, ε0 = Permittivity of free space ε0 = 8.8562 x 10-12 F/m K= Dielectric constant of material between plates (see table) Dielectric constants Dielectric K Dielectric K Vacuum 1.0000 Quartz, fused 3.8 Air 1.0006 Wood, maple 4.4 PTFE, Teflon 2.0 Glass 4.9-7.5 Mineral oil 2.0 Castor oil 5.0 Polypropylene 2.20-2.28 Wood, birch 5.2 ABS resin 2.4 - 3.2 Mica, muscovite 5.0-8.7 Polystyrene 2.45-4.0 Glass-bonded mica 6.3-9.3 Waxed paper 2.5 Poreclain, steatite 6.5 Transformer oil 2.5-4 Alumina Al2O3 8-10.0 Wood, oak 3.3 Water, distilled 80 Hard Rubber 2.5-4.8 Ta2O5 27.6 Silicones 3.4-4.3 Ba2TiO3 1200-1500 Bakelite 3.5-6.0 BaSrTiO3 7500 A formula for capacitance in picofarads using practical dimensions: 0.0885K(n-1) A 0.225K(n-1)A’ C= = d d’ A Where, d C = Capacitance in picofarads K = Dielectric constant A = Area of one plate in square centimeters A’ = Area of one plate in square inches d= Thickness in centimeters d’ = Thickness in inches n= Number of plates
  14. 14. 6 CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS 1.6 Inductor sizing equation N2µA L= l µ = µr µ0 r Where, l L = Inductance of coil in Henrys N = Number of turns in wire coil (straight wire = 1) µ = Permeability of core material (absolute, not relative) µr = Relative permeability, dimensionless ( µ0=1 for air) -6 µ0 = 1.26 x 10 T-m/At permeability of free space A = Area of coil in square meters = πr2 l = Average length of coil in meters Wheeler’s formulas for inductance of air core coils which follow are usefull for radio fre- quency inductors. The following formula for the inductance of a single layer air core solenoid coil is accurate to approximately 1% for 2r/l < 3. The thick coil formula is 1% accurate when the denominator terms are approximately equal. Wheeler’s spiral formula is 1% accurate for c>0.2r. While this is a ”round wire” formula, it may still be applicable to printed circuit spiral inductors at reduced accuracy. r c c r r l 2 2 Nr L= 9r + 10⋅l l 0.8N2r2 N2r2 L= L= 6r + 9⋅l + 10c 8r + 11c Where, L = Inductance of coil in microhenrys N = Number of turns of wire r = Mean radius of coil in inches l = Length of coil in inches c = Thickness of coil in inches
  15. 15. 1.7. TIME CONSTANT EQUATIONS 7 The inductance in henries of a square printed circuit inductor is given by two formulas where p=q, and p=q. L = 85⋅10-10DN5/3 L = 27⋅10-10(D8/3/p5/3)(1+R-1)5/3 p Where, Where, D D = dimension, cm D = coil dimension in cm N = number turns q N = number of turns p=q R= p/q The wire table provides ”turns per inch” for enamel magnet wire for use with the inductance formulas for coils. AWG turns/ AWG turns/ AWG turns/ AWG turns/ gauge inch gauge inch gauge inch gauge inch 10 9.6 20 29.4 30 90.5 40 282 11 10.7 21 33.1 31 101 41 327 12 12.0 22 37.0 32 113 42 378 13 13.5 23 41.3 33 127 43 421 14 15.0 24 46.3 34 143 44 471 15 16.8 25 51.7 35 158 45 523 16 18.9 26 58.0 36 175 46 581 17 21.2 27 64.9 37 198 18 23.6 28 72.7 38 224 19 26.4 29 81.6 39 248 1.7 Time constant equations 1.7.1 Value of time constant in series RC and RL circuits Time constant in seconds = RC Time constant in seconds = L/R
  16. 16. 8 CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS 1.7.2 Calculating voltage or current at specified time Universal Time Constant Formula 1 Change = (Final-Start) 1 - et/τ Where, Final = Value of calculated variable after infinite time (its ultimate value) Start = Initial value of calculated variable e = Euler’s number ( 2.7182818) t = Time in seconds τ = Time constant for circuit in seconds 1.7.3 Calculating time at specified voltage or current 1 t=τ ln Change 1- Final - Start 1.8 AC circuit equations 1.8.1 Inductive reactance XL = 2πfL Where, XL = Inductive reactance in ohms f = Frequency in hertz L = Inductance in henrys
  17. 17. 1.8. AC CIRCUIT EQUATIONS 9 1.8.2 Capacitive reactance XC = 1 2πfC Where, XC = Inductive reactance in ohms f = Frequency in hertz C = Capacitance in farads 1.8.3 Impedance in relation to R and X ZL = R + jXL ZC = R - jXC 1.8.4 Ohm’s Law for AC E = IZ I= E Z= E Z I Where, E = Voltage in volts I = Current in amperes (amps) Z = Impedance in ohms 1.8.5 Series and Parallel Impedances Zseries = Z1 + Z2 + . . . Zn 1 Zparallel = 1 1 1 Z1 + Z2 + . . . Zn NOTE: All impedances must be calculated in complex number form for these equations to work.
  18. 18. 10 CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS 1.8.6 Resonance 1 fresonant = 2π LC NOTE: This equation applies to a non-resistive LC circuit. In circuits containing resistance as well as inductance and capacitance, this equation applies only to series configurations and to parallel configurations where R is very small. 1.8.7 AC power E2 P = true power P = I2R P= R Measured in units of Watts E2 Q = reactive power Q = I2X Q= X Measured in units of Volt-Amps-Reactive (VAR) E2 S = apparent power S = I2Z S= S = IE Z Measured in units of Volt-Amps P = (IE)(power factor) S= P2 + Q2 Power factor = cos (Z phase angle)
  19. 19. 1.9. DECIBELS 11 1.9 Decibels AV(dB) AV(dB) = 20 log AV(ratio) AV(ratio) = 10 20 AI(dB) 20 AI(dB) = 20 log AI(ratio) AI(ratio) = 10 AP(dB) 10 AP(dB) = 10 log AP(ratio) AP(ratio) = 10
  20. 20. 12 CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS 1.10 Metric prefixes and unit conversions • Metric prefixes • Yotta = 1024 Symbol: Y • Zetta = 1021 Symbol: Z • Exa = 1018 Symbol: E • Peta = 1015 Symbol: P • Tera = 1012 Symbol: T • Giga = 109 Symbol: G • Mega = 106 Symbol: M • Kilo = 103 Symbol: k • Hecto = 102 Symbol: h • Deca = 101 Symbol: da • Deci = 10−1 Symbol: d • Centi = 10−2 Symbol: c • Milli = 10−3 Symbol: m • Micro = 10−6 Symbol: µ • Nano = 10−9 Symbol: n • Pico = 10−12 Symbol: p • Femto = 10−15 Symbol: f • Atto = 10−18 Symbol: a • Zepto = 10−21 Symbol: z • Yocto = 10−24 Symbol: y METRIC PREFIX SCALE T G M k m µ n p tera giga mega kilo (none) milli micro nano pico 1012 109 106 103 100 10-3 10-6 10-9 10-12 102 101 10-1 10-2 hecto deca deci centi h da d c
  21. 21. 1.10. METRIC PREFIXES AND UNIT CONVERSIONS 13 • Conversion factors for temperature • o F = (o C)(9/5) + 32 • o C = (o F - 32)(5/9) • o R = o F + 459.67 • o K = o C + 273.15 Conversion equivalencies for volume 1 US gallon (gal) = 231.0 cubic inches (in3 ) = 4 quarts (qt) = 8 pints (pt) = 128 fluid ounces (fl. oz.) = 3.7854 liters (l) 1 Imperial gallon (gal) = 160 fluid ounces (fl. oz.) = 4.546 liters (l) Conversion equivalencies for distance 1 inch (in) = 2.540000 centimeter (cm) Conversion equivalencies for velocity 1 mile per hour (mi/h) = 88 feet per minute (ft/m) = 1.46667 feet per second (ft/s) = 1.60934 kilometer per hour (km/h) = 0.44704 meter per second (m/s) = 0.868976 knot (knot – international) Conversion equivalencies for weight 1 pound (lb) = 16 ounces (oz) = 0.45359 kilogram (kg) Conversion equivalencies for force 1 pound-force (lbf) = 4.44822 newton (N) Acceleration of gravity (free fall), Earth standard 9.806650 meters per second per second (m/s2 ) = 32.1740 feet per second per sec- ond (ft/s2 ) Conversion equivalencies for area 1 acre = 43560 square feet (ft2 ) = 4840 square yards (yd2 ) = 4046.86 square meters (m2 )
  22. 22. 14 CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS Conversion equivalencies for pressure 1 pound per square inch (psi) = 2.03603 inches of mercury (in. Hg) = 27.6807 inches of water (in. W.C.) = 6894.757 pascals (Pa) = 0.0680460 atmospheres (Atm) = 0.0689476 bar (bar) Conversion equivalencies for energy or work 1 british thermal unit (BTU – ”International Table”) = 251.996 calories (cal – ”International Table”) = 1055.06 joules (J) = 1055.06 watt-seconds (W-s) = 0.293071 watt-hour (W-hr) = 1.05506 x 1010 ergs (erg) = 778.169 foot-pound-force (ft-lbf) Conversion equivalencies for power 1 horsepower (hp – 550 ft-lbf/s) = 745.7 watts (W) = 2544.43 british thermal units per hour (BTU/hr) = 0.0760181 boiler horsepower (hp – boiler) Conversion equivalencies for motor torque Newton-meter Gram-centimeter Pound-inch Pound-foot Ounce-inch (n-m) (g-cm) (lb-in) (lb-ft) (oz-in) n-m 1 1020 8.85 0.738 141.6 -6 -3 -6 g-cm 981 x 10 1 8.68 x 10 723 x 10 0.139 lb-in 0.113 115 1 0.0833 16 lb-ft 1.36 1383 12 1 192 -3 -3 oz-in 7.062 x 10 7.20 0.0625 5.21 x 10 1 Locate the row corresponding to known unit of torque along the left of the table. Multiply by the factor under the column for the desired units. For example, to convert 2 oz-in torque to n-m, locate oz-in row at table left. Locate 7.062 x 10−3 at intersection of desired n-m units column. Multiply 2 oz-in x (7.062 x 10−3 ) = 14.12 x 10−3 n-m. Converting between units is easy if you have a set of equivalencies to work with. Suppose we wanted to convert an energy quantity of 2500 calories into watt-hours. What we would need to do is find a set of equivalent figures for those units. In our reference here, we see that 251.996 calories is physically equal to 0.293071 watt hour. To convert from calories into watt-hours, we must form a ”unity fraction” with these physically equal figures (a fraction composed of different figures and different units, the numerator and denominator being physically equal to one another), placing the desired unit in the numerator and the initial unit in the denominator, and then multiply our initial value of calories by that fraction. Since both terms of the ”unity fraction” are physically equal to one another, the fraction as a whole has a physical value of 1, and so does not change the true value of any figure when multiplied by it. When units are canceled, however, there will be a change in units.
  23. 23. 1.10. METRIC PREFIXES AND UNIT CONVERSIONS 15 For example, 2500 calories multiplied by the unity fraction of (0.293071 w-hr / 251.996 cal) = 2.9075 watt-hours. Original figure 2500 calories 0.293071 watt-hour "Unity fraction" 251.996 calories . . . cancelling units . . . 2500 calories 0.293071 watt-hour 1 251.996 calories Converted figure 2.9075 watt-hours The ”unity fraction” approach to unit conversion may be extended beyond single steps. Sup- pose we wanted to convert a fluid flow measurement of 175 gallons per hour into liters per day. We have two units to convert here: gallons into liters, and hours into days. Remember that the word ”per” in mathematics means ”divided by,” so our initial figure of 175 gallons per hour means 175 gallons divided by hours. Expressing our original figure as such a fraction, we multiply it by the necessary unity fractions to convert gallons to liters (3.7854 liters = 1 gal- lon), and hours to days (1 day = 24 hours). The units must be arranged in the unity fraction in such a way that undesired units cancel each other out above and below fraction bars. For this problem it means using a gallons-to-liters unity fraction of (3.7854 liters / 1 gallon) and a hours-to-days unity fraction of (24 hours / 1 day):
  24. 24. 16 CHAPTER 1. USEFUL EQUATIONS AND CONVERSION FACTORS Original figure 175 gallons/hour 3.7854 liters "Unity fraction" 1 gallon 24 hours "Unity fraction" 1 day . . . cancelling units . . . 175 gallons 3.7854 liters 24 hours 1 hour 1 gallon 1 day Converted figure 15,898.68 liters/day Our final (converted) answer is 15898.68 liters per day. 1.11 Data Conversion factors were found in the 78th edition of the CRC Handbook of Chemistry and Physics, and the 3rd edition of Bela Liptak’s Instrument Engineers’ Handbook – Process Mea- surement and Analysis. 1.12 Contributors Contributors to this chapter are listed in chronological order of their contributions, from most recent to first. See Appendix 2 (Contributor List) for dates and contact information. Gerald Gardner (January 2003): Addition of Imperial gallons conversion.
  25. 25. Chapter 2 COLOR CODES Contents 2.1 Resistor Color Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.1 Example #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.2 Example #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.3 Example #3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.4 Example #4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.5 Example #5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.6 Example #6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Wiring Color Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Components and wires are coded are with colors to identify their value and function. 2.1 Resistor Color Codes Components and wires are coded are with colors to identify their value and function. 17
  26. 26. 18 CHAPTER 2. COLOR CODES Color Digit Multiplier Tolerance (%) Black 0 100 (1) Brown 1 101 1 2 Red 2 10 2 Orange 3 103 Yellow 4 104 Green 5 105 0.5 6 Blue 6 10 0.25 Violet 7 107 0.1 8 Grey 8 10 White 9 109 Gold 10-1 5 Silver 10-2 10 (none) 20 The colors brown, red, green, blue, and violet are used as tolerance codes on 5-band resistors only. All 5-band resistors use a colored tolerance band. The blank (20%) ”band” is only used with the ”4-band” code (3 colored bands + a blank ”band”). Digit Digit Multiplier Tolerance 4-band code Digit Digit Digit Multiplier Tolerance 5-band code
  27. 27. 2.1. RESISTOR COLOR CODES 19 2.1.1 Example #1 A resistor colored Yellow-Violet-Orange-Gold would be 47 kΩ with a tolerance of +/- 5%. 2.1.2 Example #2 A resistor colored Green-Red-Gold-Silver would be 5.2 Ω with a tolerance of +/- 10%. 2.1.3 Example #3 A resistor colored White-Violet-Black would be 97 Ω with a tolerance of +/- 20%. When you see only three color bands on a resistor, you know that it is actually a 4-band code with a blank (20%) tolerance band. 2.1.4 Example #4 A resistor colored Orange-Orange-Black-Brown-Violet would be 3.3 kΩ with a tolerance of +/- 0.1%. 2.1.5 Example #5 A resistor colored Brown-Green-Grey-Silver-Red would be 1.58 Ω with a tolerance of +/- 2%. 2.1.6 Example #6 A resistor colored Blue-Brown-Green-Silver-Blue would be 6.15 Ω with a tolerance of +/- 0.25%.
  28. 28. 20 CHAPTER 2. COLOR CODES 2.2 Wiring Color Codes Wiring for AC and DC power distribution branch circuits are color coded for identification of individual wires. In some jurisdictions all wire colors are specified in legal documents. In other jurisdictions, only a few conductor colors are so codified. In that case, local custom dictates the “optional” wire colors. IEC, AC: Most of Europe abides by IEC (International Electrotechnical Commission) wiring color codes for AC branch circuits. These are listed in Table 2.1. The older color codes in the table reflect the previous style which did not account for proper phase rotation. The protective ground wire (listed as green-yellow) is green with yellow stripe. Table 2.1: IEC (most of Europe) AC power circuit wiring color codes. Function label Color, IEC Color, old IEC Protective earth PE green-yellow green-yellow Neutral N blue blue Line, single phase L brown brown or black Line, 3-phase L1 brown brown or black Line, 3-phase L2 black brown or black Line, 3-phase L3 grey brown or black UK, AC: The United Kingdom now follows the IEC AC wiring color codes. Table 2.2 lists these along with the obsolete domestic color codes. For adding new colored wiring to existing old colored wiring see Cook. [1] Table 2.2: UK AC power circuit wiring color codes. Function label Color, IEC Old UK color Protective earth PE green-yellow green-yellow Neutral N blue black Line, single phase L brown red Line, 3-phase L1 brown red Line, 3-phase L2 black yellow Line, 3-phase L3 grey blue US, AC:The US National Electrical Code only mandates white (or grey) for the neutral power conductor and bare copper, green, or green with yellow stripe for the protective ground. In principle any other colors except these may be used for the power conductors. The colors adopted as local practice are shown in Table 2.3. Black, red, and blue are used for 208 VAC three-phase; brown, orange and yellow are used for 480 VAC. Conductors larger than #6 AWG are only available in black and are color taped at the ends. Canada: Canadian wiring is governed by the CEC (Canadian Electric Code). See Table 2.4. The protective ground is green or green with yellow stripe. The neutral is white, the hot (live or active) single phase wires are black , and red in the case of a second active. Three-phase lines are red, black, and blue.
  29. 29. 2.2. WIRING COLOR CODES 21 Table 2.3: US AC power circuit wiring color codes. Function label Color, common Color, alternative Protective ground PG bare, green, or green-yellow green Neutral N white grey Line, single phase L black or red (2nd hot) Line, 3-phase L1 black brown Line, 3-phase L2 red orange Line, 3-phase L3 blue yellow Table 2.4: Canada AC power circuit wiring color codes. Function label Color, common Protective ground PG green or green-yellow Neutral N white Line, single phase L black or red (2nd hot) Line, 3-phase L1 red Line, 3-phase L2 black Line, 3-phase L3 blue IEC, DC: DC power installations, for example, solar power and computer data centers, use color coding which follows the AC standards. The IEC color standard for DC power cables is listed in Table 2.5, adapted from Table 2, Cook. [1] Table 2.5: IEC DC power circuit wiring color codes. Function label Color Protective earth PE green-yellow 2-wire unearthed DC Power System Positive L+ brown Negative L- grey 2-wire earthed DC Power System Positive (of a negative earthed) circuit L+ brown Negative (of a negative earthed) circuit M blue Positive (of a positive earthed) circuit M blue Negative (of a positive earthed) circuit L- grey 3-wire earthed DC Power System Positive L+ brown Mid-wire M blue Negative L- grey US DC power: The US National Electrical Code (for both AC and DC) mandates that the grounded neutral conductor of a power system be white or grey. The protective ground must be bare, green or green-yellow striped. Hot (active) wires may be any other colors except these. However, common practice (per local electrical inspectors) is for the first hot (live or active) wire to be black and the second hot to be red. The recommendations in Table 2.6 are
  30. 30. 22 CHAPTER 2. COLOR CODES by Wiles. [2] He makes no recommendation for ungrounded power system colors. Usage of the ungrounded system is discouraged for safety. However, red (+) and black (-) follows the coloring of the grounded systems in the table. Table 2.6: US recommended DC power circuit wiring color codes. Function label Color Protective ground PG bare, green, or green-yellow 2-wire ungrounded DC Power System Positive L+ no recommendation (red) Negative L- no recommendation (black) 2-wire grounded DC Power System Positive (of a negative grounded) circuit L+ red Negative (of a negative grounded) circuit N white Positive (of a positive grounded) circuit N white Negative (of a positive grounded) circuit L- black 3-wire grounded DC Power System Positive L+ red Mid-wire (center tap) N white Negative L- black Bibliography [1] Paul Cook, “Harmonised colours and alphanumeric marking”, IEE Wiring Matters, Spring 2004 at http://www.iee.org/Publish/WireRegs/IEE Harmonized colours.pdf [2] John Wiles, “Photovoltaic Power Systems and the National Electrical Code: Suggested Practices”, Southwest Technology Development Institute, New Mexico State University, March 2001 at http://www.re.sandia.gov/en/ti/tu/Copy%20of%20NEC2000.pdf
  31. 31. Chapter 3 CONDUCTOR AND INSULATOR TABLES Contents 3.1 Copper wire gage table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Copper wire ampacity table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Coefficients of specific resistance . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4 Temperature coefficients of resistance . . . . . . . . . . . . . . . . . . . . . 26 3.5 Critical temperatures for superconductors . . . . . . . . . . . . . . . . . . 26 3.6 Dielectric strengths for insulators . . . . . . . . . . . . . . . . . . . . . . . . 27 3.7 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1 Copper wire gage table Soild copper wire table: Size Diameter Cross-sectional area Weight AWG inches cir. mils sq. inches lb/1000 ft ================================================================ 4/0 -------- 0.4600 ------- 211,600 ------ 0.1662 ------ 640.5 3/0 -------- 0.4096 ------- 167,800 ------ 0.1318 ------ 507.9 2/0 -------- 0.3648 ------- 133,100 ------ 0.1045 ------ 402.8 1/0 -------- 0.3249 ------- 105,500 ----- 0.08289 ------ 319.5 1 ---------- 0.2893 ------- 83,690 ------ 0.06573 ------ 253.5 2 ---------- 0.2576 ------- 66,370 ------ 0.05213 ------ 200.9 3 ---------- 0.2294 ------- 52,630 ------ 0.04134 ------ 159.3 4 ---------- 0.2043 ------- 41,740 ------ 0.03278 ------ 126.4 5 ---------- 0.1819 ------- 33,100 ------ 0.02600 ------ 100.2 6 ---------- 0.1620 ------- 26,250 ------ 0.02062 ------ 79.46 23
  32. 32. 24 CHAPTER 3. CONDUCTOR AND INSULATOR TABLES 7 ---------- 0.1443 ------- 20,820 ------ 0.01635 ------ 63.02 8 ---------- 0.1285 ------- 16,510 ------ 0.01297 ------ 49.97 9 ---------- 0.1144 ------- 13,090 ------ 0.01028 ------ 39.63 10 --------- 0.1019 ------- 10,380 ------ 0.008155 ----- 31.43 11 --------- 0.09074 ------- 8,234 ------ 0.006467 ----- 24.92 12 --------- 0.08081 ------- 6,530 ------ 0.005129 ----- 19.77 13 --------- 0.07196 ------- 5,178 ------ 0.004067 ----- 15.68 14 --------- 0.06408 ------- 4,107 ------ 0.003225 ----- 12.43 15 --------- 0.05707 ------- 3,257 ------ 0.002558 ----- 9.858 16 --------- 0.05082 ------- 2,583 ------ 0.002028 ----- 7.818 17 --------- 0.04526 ------- 2,048 ------ 0.001609 ----- 6.200 18 --------- 0.04030 ------- 1,624 ------ 0.001276 ----- 4.917 19 --------- 0.03589 ------- 1,288 ------ 0.001012 ----- 3.899 20 --------- 0.03196 ------- 1,022 ----- 0.0008023 ----- 3.092 21 --------- 0.02846 ------- 810.1 ----- 0.0006363 ----- 2.452 22 --------- 0.02535 ------- 642.5 ----- 0.0005046 ----- 1.945 23 --------- 0.02257 ------- 509.5 ----- 0.0004001 ----- 1.542 24 --------- 0.02010 ------- 404.0 ----- 0.0003173 ----- 1.233 25 --------- 0.01790 ------- 320.4 ----- 0.0002517 ----- 0.9699 26 --------- 0.01594 ------- 254.1 ----- 0.0001996 ----- 0.7692 27 --------- 0.01420 ------- 201.5 ----- 0.0001583 ----- 0.6100 28 --------- 0.01264 ------- 159.8 ----- 0.0001255 ----- 0.4837 29 --------- 0.01126 ------- 126.7 ----- 0.00009954 ---- 0.3836 30 --------- 0.01003 ------- 100.5 ----- 0.00007894 ---- 0.3042 31 -------- 0.008928 ------- 79.70 ----- 0.00006260 ---- 0.2413 32 -------- 0.007950 ------- 63.21 ----- 0.00004964 ---- 0.1913 33 -------- 0.007080 ------- 50.13 ----- 0.00003937 ---- 0.1517 34 -------- 0.006305 ------- 39.75 ----- 0.00003122 ---- 0.1203 35 -------- 0.005615 ------- 31.52 ----- 0.00002476 --- 0.09542 36 -------- 0.005000 ------- 25.00 ----- 0.00001963 --- 0.07567 37 -------- 0.004453 ------- 19.83 ----- 0.00001557 --- 0.06001 38 -------- 0.003965 ------- 15.72 ----- 0.00001235 --- 0.04759 39 -------- 0.003531 ------- 12.47 ---- 0.000009793 --- 0.03774 40 -------- 0.003145 ------- 9.888 ---- 0.000007766 --- 0.02993 41 -------- 0.002800 ------- 7.842 ---- 0.000006159 --- 0.02374 42 -------- 0.002494 ------- 6.219 ---- 0.000004884 --- 0.01882 43 -------- 0.002221 ------- 4.932 ---- 0.000003873 --- 0.01493 44 -------- 0.001978 ------- 3.911 ---- 0.000003072 --- 0.01184 3.2 Copper wire ampacity table Ampacities of copper wire, in free air at 30o C: ======================================================== | INSULATION TYPE: | | RUW, T THW, THWN FEP, FEPB |
  33. 33. 3.3. COEFFICIENTS OF SPECIFIC RESISTANCE 25 | TW RUH THHN, XHHW | ======================================================== Size Current Rating Current Rating Current Rating AWG @ 60 degrees C @ 75 degrees C @ 90 degrees C ======================================================== 20 -------- *9 ----------------------------- *12.5 18 -------- *13 ------------------------------ 18 16 -------- *18 ------------------------------ 24 14 --------- 25 ------------- 30 ------------- 35 12 --------- 30 ------------- 35 ------------- 40 10 --------- 40 ------------- 50 ------------- 55 8 ---------- 60 ------------- 70 ------------- 80 6 ---------- 80 ------------- 95 ------------ 105 4 --------- 105 ------------ 125 ------------ 140 2 --------- 140 ------------ 170 ------------ 190 1 --------- 165 ------------ 195 ------------ 220 1/0 ------- 195 ------------ 230 ------------ 260 2/0 ------- 225 ------------ 265 ------------ 300 3/0 ------- 260 ------------ 310 ------------ 350 4/0 ------- 300 ------------ 360 ------------ 405 * = estimated values; normally, wire gages this small are not manufactured with these insulation types. 3.3 Coefficients of specific resistance Specific resistance at 20o C: Material Element/Alloy (ohm-cmil/ft) (ohm-cm·10−6 ) ==================================================================== Nichrome ------- Alloy ---------------- 675 ------------- 112.2 Nichrome V ----- Alloy ---------------- 650 ------------- 108.1 Manganin ------- Alloy ---------------- 290 ------------- 48.21 Constantan ----- Alloy ---------------- 272.97 ---------- 45.38 Steel* --------- Alloy ---------------- 100 ------------- 16.62 Platinum ------ Element --------------- 63.16 ----------- 10.5 Iron ---------- Element --------------- 57.81 ----------- 9.61 Nickel -------- Element --------------- 41.69 ----------- 6.93 Zinc ---------- Element --------------- 35.49 ----------- 5.90 Molybdenum ---- Element --------------- 32.12 ----------- 5.34 Tungsten ------ Element --------------- 31.76 ----------- 5.28 Aluminum ------ Element --------------- 15.94 ----------- 2.650 Gold ---------- Element --------------- 13.32 ----------- 2.214 Copper -------- Element --------------- 10.09 ----------- 1.678 Silver -------- Element --------------- 9.546 ----------- 1.587 * = Steel alloy at 99.5 percent iron, 0.5 percent carbon.
  34. 34. 26 CHAPTER 3. CONDUCTOR AND INSULATOR TABLES 3.4 Temperature coefficients of resistance Temperature coefficient (α) per degree C: Material Element/Alloy Temp. coefficient ===================================================== Nickel -------- Element --------------- 0.005866 Iron ---------- Element --------------- 0.005671 Molybdenum ---- Element --------------- 0.004579 Tungsten ------ Element --------------- 0.004403 Aluminum ------ Element --------------- 0.004308 Copper -------- Element --------------- 0.004041 Silver -------- Element --------------- 0.003819 Platinum ------ Element --------------- 0.003729 Gold ---------- Element --------------- 0.003715 Zinc ---------- Element --------------- 0.003847 Steel* --------- Alloy ---------------- 0.003 Nichrome ------- Alloy ---------------- 0.00017 Nichrome V ----- Alloy ---------------- 0.00013 Manganin ------- Alloy ------------ +/- 0.000015 Constantan ----- Alloy --------------- -0.000074 * = Steel alloy at 99.5 percent iron, 0.5 percent carbon 3.5 Critical temperatures for superconductors Critical temperatures given in Kelvins Material Element/Alloy Critical temperature(K) ======================================================= Aluminum -------- Element --------------- 1.20 Cadmium --------- Element --------------- 0.56 Lead ------------ Element --------------- 7.2 Mercury --------- Element --------------- 4.16 Niobium --------- Element --------------- 8.70 Thorium --------- Element --------------- 1.37 Tin ------------- Element --------------- 3.72 Titanium -------- Element --------------- 0.39 Uranium --------- ELement --------------- 1.0 Zinc ------------ Element --------------- 0.91 Niobium/Tin ------ Alloy ---------------- 18.1 Cupric sulphide - Compound -------------- 1.6
  35. 35. 3.6. DIELECTRIC STRENGTHS FOR INSULATORS 27 Critical temperatures, high temperature superconuctors ni Kelvins Material Critical temperature(K) ======================================================= HgBa2 Ca2 Cu3 O8+d ---------------- 150 (23.5 GPa pressure) HgBa2 Ca2 Cu3 O8+d ---------------- 133 Tl2 Ba2 Ca2 Cu3 O10 ---------------- 125 YBa2 Cu3 O7 ---------------------- 90 La1.85 Sr0.15 CuO4 ----------------- 40 Cs3 C60 ------------------------- 40 (15 Kbar pressure) Ba0.6 K0.4 BiO3 ------------------- 30 Nd1.85 Ce0.15 CuO4 ----------------- 22 K3 C60 -------------------------- 19 PbMo6 S8 ------------------------ 12.6 Note: all critical temperatures given at zero magnetic field strength. 3.6 Dielectric strengths for insulators Dielectric strength in kilovolts per inch (kV/in): Material* Dielectric strength ========================================= Vacuum --------------------- 20 Air ------------------------ 20 to 75 Porcelain ------------------ 40 to 200 Paraffin Wax --------------- 200 to 300 Transformer Oil ------------ 400 Bakelite ------------------- 300 to 550 Rubber --------------------- 450 to 700 Shellac -------------------- 900 Paper ---------------------- 1250 Teflon --------------------- 1500 Glass ---------------------- 2000 to 3000 Mica ----------------------- 5000 * = Materials listed are specially prepared for electrical use 3.7 Data Tables of specific resistance and temperature coefficient of resistance for elemental materials (not alloys) were derived from figures found in the 78th edition of the CRC Handbook of Chem- istry and Physics. Superconductivity data from Collier’s Encyclopedia (volume 21, 1968, page 640).
  36. 36. 28 CHAPTER 3. CONDUCTOR AND INSULATOR TABLES
  37. 37. Chapter 4 ALGEBRA REFERENCE Contents 4.1 Basic identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Arithmetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.1 The associative property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.2 The commutative property . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.3 The distributive property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Properties of exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.4 Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.4.1 Definition of a radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.4.2 Properties of radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.5 Important constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.5.1 Euler’s number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.5.2 Pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.6 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.6.1 Definition of a logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.6.2 Properties of logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.7 Factoring equivalencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.8 The quadratic formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.9 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.9.1 Arithmetic sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.9.2 Geometric sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.10 Factorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.10.1 Definition of a factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.10.2 Strange factorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.11 Solving simultaneous equations . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.11.1 Substitution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.11.2 Addition method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.12 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 29
  38. 38. 30 CHAPTER 4. ALGEBRA REFERENCE 4.1 Basic identities a+0=a 1a = a 0a = 0 a =a 0 =0 a =1 1 a a a = undefined 0 Note: while division by zero is popularly thought to be equal to infinity, this is not techni- cally true. In some practical applications it may be helpful to think the result of such a fraction approaching positive infinity as a positive denominator approaches zero (imagine calculating current I=E/R in a circuit with resistance approaching zero – current would approach infinity), but the actual fraction of anything divided by zero is undefined in the scope of either real or complex numbers. 4.2 Arithmetic properties 4.2.1 The associative property In addition and multiplication, terms may be arbitrarily associated with each other through the use of parentheses: a + (b + c) = (a + b) + c a(bc) = (ab)c 4.2.2 The commutative property In addition and multiplication, terms may be arbitrarily interchanged, or commutated: a+b=b+a ab=ba 4.2.3 The distributive property a(b + c) = ab + ac 4.3 Properties of exponents aman = am+n (ab)m = ambm am (am)n = amn = am-n an
  39. 39. 4.4. RADICALS 31 4.4 Radicals 4.4.1 Definition of a radical When people talk of a ”square root,” they’re referring to a radical with a root of 2. This is mathematically equivalent to a number raised to the power of 1/2. This equivalence is useful to know when using a calculator to determine a strange root. Suppose for example you needed to find the fourth root of a number, but your calculator lacks a ”4th root” button or function. If it has a yx function (which any scientific calculator should have), you can find the fourth root by raising that number to the 1/4 power, or x0.25 . x a = a1/x It is important to remember that when solving for an even root (square root, fourth root, etc.) of any number, there are two valid answers. For example, most people know that the square root of nine is three, but negative three is also a valid answer, since (-3)2 = 9 just as 32 = 9. 4.4.2 Properties of radicals x x x a =a ax = a x x x ab = a b x x a a = x b b 4.5 Important constants 4.5.1 Euler’s number Euler’s constant is an important value for exponential functions, especially scientific applica- tions involving decay (such as the decay of a radioactive substance). It is especially important in calculus due to its uniquely self-similar properties of integration and differentiation. e approximately equals: 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69996
  40. 40. 32 CHAPTER 4. ALGEBRA REFERENCE e= 1 k! k=0 1 1 1 1 ... 1 0! + 1! + 2! + 3! + n! 4.5.2 Pi Pi (π) is defined as the ratio of a circle’s circumference to its diameter. Pi approximately equals: 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37511 Note: For both Euler’s constant (e) and pi (π), the spaces shown between each set of five digits have no mathematical significance. They are placed there just to make it easier for your eyes to ”piece” the number into five-digit groups when manually copying. 4.6 Logarithms 4.6.1 Definition of a logarithm If: by = x Then: logb x = y Where, b = "Base" of the logarithm ”log” denotes a common logarithm (base = 10), while ”ln” denotes a natural logarithm (base = e).
  41. 41. 4.7. FACTORING EQUIVALENCIES 33 4.6.2 Properties of logarithms (log a) + (log b) = log ab (log a) - (log b) = log a b log am = (m)(log a) a(log m) = m These properties of logarithms come in handy for performing complex multiplication and division operations. They are an example of something called a transform function, whereby one type of mathematical operation is transformed into another type of mathematical operation that is simpler to solve. Using a table of logarithm figures, one can multiply or divide numbers by adding or subtracting their logarithms, respectively. then looking up that logarithm figure in the table and seeing what the final product or quotient is. Slide rules work on this principle of logarithms by performing multiplication and division through addition and subtraction of distances on the slide. Slide rule Cursor Slide Numerical quantities are represented by the positioning of the slide. Marks on a slide rule’s scales are spaced in a logarithmic fashion, so that a linear posi- tioning of the scale or cursor results in a nonlinear indication as read on the scale(s). Adding or subtracting lengths on these logarithmic scales results in an indication equivalent to the product or quotient, respectively, of those lengths. Most slide rules were also equipped with special scales for trigonometric functions, powers, roots, and other useful arithmetic functions. 4.7 Factoring equivalencies x2 - y2 = (x+y)(x-y) x3 - y3 = (x-y)(x2 + xy + y2)
  42. 42. 34 CHAPTER 4. ALGEBRA REFERENCE 4.8 The quadratic formula For a polynomial expression in the form of: ax2 + bx + c = 0 + x = -b - b2 - 4ac 2a 4.9 Sequences 4.9.1 Arithmetic sequences An arithmetic sequence is a series of numbers obtained by adding (or subtracting) the same value with each step. A child’s counting sequence (1, 2, 3, 4, . . .) is a simple arithmetic sequence, where the common difference is 1: that is, each adjacent number in the sequence differs by a value of one. An arithmetic sequence counting only even numbers (2, 4, 6, 8, . . .) or only odd numbers (1, 3, 5, 7, 9, . . .) would have a common difference of 2. In the standard notation of sequences, a lower-case letter ”a” represents an element (a single number) in the sequence. The term ”an ” refers to the element at the nth step in the sequence. For example, ”a3 ” in an even-counting (common difference = 2) arithmetic sequence starting at 2 would be the number 6, ”a” representing 4 and ”a1 ” representing the starting point of the sequence (given in this example as 2). A capital letter ”A” represents the sum of an arithmetic sequence. For instance, in the same even-counting sequence starting at 2, A4 is equal to the sum of all elements from a1 through a4 , which of course would be 2 + 4 + 6 + 8, or 20. an = an-1 + d an = a1 + d(n-1) Where: d = The "common difference" Example of an arithmetic sequence: -7, -3, 1, 5, 9, 13, 17, 21, 25 . . . An = a1 + a2 + . . . an An = n (a1 + an) 2
  43. 43. 4.10. FACTORIALS 35 4.9.2 Geometric sequences A geometric sequence, on the other hand, is a series of numbers obtained by multiplying (or dividing) by the same value with each step. A binary place-weight sequence (1, 2, 4, 8, 16, 32, 64, . . .) is a simple geometric sequence, where the common ratio is 2: that is, each adjacent number in the sequence differs by a factor of two. an = r(an-1) an = a1(rn-1) Where: r = The "common ratio" Example of a geometric sequence: 3, 12, 48, 192, 768, 3072 . . . An = a1 + a2 + . . . an a1(1 - rn) An = 1-r 4.10 Factorials 4.10.1 Definition of a factorial Denoted by the symbol ”!” after an integer; the product of that integer and all integers in descent to 1. Example of a factorial: 5! = 5 x 4 x 3 x 2 x 1 5! = 120 4.10.2 Strange factorials 0! = 1 1! = 1 4.11 Solving simultaneous equations The terms simultaneous equations and systems of equations refer to conditions where two or more unknown variables are related to each other through an equal number of equations. Consider the following example:
  44. 44. 36 CHAPTER 4. ALGEBRA REFERENCE x + y = 24 2x - y = -6 For this set of equations, there is but a single combination of values for x and y that will satisfy both. Either equation, considered separately, has an infinitude of valid (x,y) solutions, but together there is only one. Plotted on a graph, this condition becomes obvious: (6,18) x + y = 24 2x - y = -6 Each line is actually a continuum of points representing possible x and y solution pairs for each equation. Each equation, separately, has an infinite number of ordered pair (x,y) solu- tions. There is only one point where the two linear functions x + y = 24 and 2x - y = -6 intersect (where one of their many independent solutions happen to work for both equations), and that is where x is equal to a value of 6 and y is equal to a value of 18. Usually, though, graphing is not a very efficient way to determine the simultaneous solution set for two or more equations. It is especially impractical for systems of three or more variables. In a three-variable system, for example, the solution would be found by the point intersection of three planes in a three-dimensional coordinate space – not an easy scenario to visualize. 4.11.1 Substitution method Several algebraic techniques exist to solve simultaneous equations. Perhaps the easiest to comprehend is the substitution method. Take, for instance, our two-variable example problem: x + y = 24 2x - y = -6 In the substitution method, we manipulate one of the equations such that one variable is defined in terms of the other:
  45. 45. 4.11. SOLVING SIMULTANEOUS EQUATIONS 37 x + y = 24 y = 24 - x Defining y in terms of x Then, we take this new definition of one variable and substitute it for the same variable in the other equation. In this case, we take the definition of y, which is 24 - x and substitute this for the y term found in the other equation: y = 24 - x substitute 2x - y = -6 2x - (24 - x) = -6 Now that we have an equation with just a single variable (x), we can solve it using ”normal” algebraic techniques: 2x - (24 - x) = -6 Distributive property 2x - 24 + x = -6 Combining like terms 3x -24 = -6 Adding 24 to each side 3x = 18 Dividing both sides by 3 x=6 Now that x is known, we can plug this value into any of the original equations and obtain a value for y. Or, to save us some work, we can plug this value (6) into the equation we just generated to define y in terms of x, being that it is already in a form to solve for y:
  46. 46. 38 CHAPTER 4. ALGEBRA REFERENCE x=6 substitute y = 24 - x y = 24 - 6 y = 18 Applying the substitution method to systems of three or more variables involves a similar pattern, only with more work involved. This is generally true for any method of solution: the number of steps required for obtaining solutions increases rapidly with each additional variable in the system. To solve for three unknown variables, we need at least three equations. Consider this example: x - y + z = 10 3x + y + 2z = 34 -5x + 2y - z = -14 Being that the first equation has the simplest coefficients (1, -1, and 1, for x, y, and z, respectively), it seems logical to use it to develop a definition of one variable in terms of the other two. In this example, I’ll solve for x in terms of y and z: x - y + z = 10 Adding y and subtracting z from both sides x = y - z + 10 Now, we can substitute this definition of x where x appears in the other two equations: x = y - z + 10 x = y - z + 10 substitute substitute 3x + y + 2z = 34 -5x + 2y - z = -14 3(y - z + 10) + y + 2z = 34 -5(y - z + 10) + 2y - z = -14 Reducing these two equations to their simplest forms:
  47. 47. 4.11. SOLVING SIMULTANEOUS EQUATIONS 39 3(y - z + 10) + y + 2z = 34 -5(y - z + 10) + 2y - z = -14 Distributive property 3y - 3z + 30 + y + 2z = 34 -5y + 5z - 50 + 2y - z = -14 Combining like terms 4y - z + 30 = 34 -3y + 4z - 50 = -14 Moving constant values to right of the "=" sign 4y - z = 4 -3y + 4z = 36 So far, our efforts have reduced the system from three variables in three equations to two variables in two equations. Now, we can apply the substitution technique again to the two equations 4y - z = 4 and -3y + 4z = 36 to solve for either y or z. First, I’ll manipulate the first equation to define z in terms of y: 4y - z = 4 Adding z to both sides; subtracting 4 from both sides z = 4y - 4 Next, we’ll substitute this definition of z in terms of y where we see z in the other equation: z = 4y - 4 substitute -3y + 4z = 36 -3y + 4(4y - 4) = 36 Distributive property -3y + 16y - 16 = 36 Combining like terms 13y - 16 = 36 Adding 16 to both sides 13y = 52 Dividing both sides by 13 y=4 Now that y is a known value, we can plug it into the equation defining z in terms of y and
  48. 48. 40 CHAPTER 4. ALGEBRA REFERENCE obtain a figure for z: y=4 substitute z = 4y - 4 z = 16 - 4 z = 12 Now, with values for y and z known, we can plug these into the equation where we defined x in terms of y and z, to obtain a value for x: y=4 substitute z = 12 substitute x = y - z + 10 x = 4 - 12 + 10 x=2 In closing, we’ve found values for x, y, and z of 2, 4, and 12, respectively, that satisfy all three equations. 4.11.2 Addition method While the substitution method may be the easiest to grasp on a conceptual level, there are other methods of solution available to us. One such method is the so-called addition method, whereby equations are added to one another for the purpose of canceling variable terms. Let’s take our two-variable system used to demonstrate the substitution method: x + y = 24 2x - y = -6 One of the most-used rules of algebra is that you may perform any arithmetic operation you wish to an equation so long as you do it equally to both sides. With reference to addition, this means we may add any quantity we wish to both sides of an equation – so long as its the same quantity – without altering the truth of the equation. An option we have, then, is to add the corresponding sides of the equations together to form a new equation. Since each equation is an expression of equality (the same quantity on either
  49. 49. 4.11. SOLVING SIMULTANEOUS EQUATIONS 41 side of the = sign), adding the left-hand side of one equation to the left-hand side of the other equation is valid so long as we add the two equations’ right-hand sides together as well. In our example equation set, for instance, we may add x + y to 2x - y, and add 24 and -6 together as well to form a new equation. What benefit does this hold for us? Examine what happens when we do this to our example equation set: x + y = 24 + 2x - y = -6 3x + 0 = 18 Because the top equation happened to contain a positive y term while the bottom equation happened to contain a negative y term, these two terms canceled each other in the process of addition, leaving no y term in the sum. What we have left is a new equation, but one with only a single unknown variable, x! This allows us to easily solve for the value of x: 3x + 0 = 18 Dropping the 0 term 3x = 18 Dividing both sides by 3 x=6 Once we have a known value for x, of course, determining y’s value is a simply matter of substitution (replacing x with the number 6) into one of the original equations. In this example, the technique of adding the equations together worked well to produce an equation with a single unknown variable. What about an example where things aren’t so simple? Consider the following equation set: 2x + 2y = 14 3x + y = 13 We could add these two equations together – this being a completely valid algebraic opera- tion – but it would not profit us in the goal of obtaining values for x and y: 2x + 2y = 14 + 3x + y = 13 5x + 3y = 27 The resulting equation still contains two unknown variables, just like the original equations do, and so we’re no further along in obtaining a solution. However, what if we could manipulate one of the equations so as to have a negative term that would cancel the respective term in the other equation when added? Then, the system would reduce to a single equation with a single unknown variable just as with the last (fortuitous) example. If we could only turn the y term in the lower equation into a - 2y term, so that when the two equations were added together, both y terms in the equations would cancel, leaving us with only an x term, this would bring us closer to a solution. Fortunately, this is not difficult to do. If we multiply each and every term of the lower equation by a -2, it will produce the result
  50. 50. 42 CHAPTER 4. ALGEBRA REFERENCE we seek: -2(3x + y) = -2(13) Distributive property -6x - 2y = -26 Now, we may add this new equation to the original, upper equation: 2x + 2y = 14 + -6x - 2y = -26 -4x + 0y = -12 Solving for x, we obtain a value of 3: -4x + 0y = -12 Dropping the 0 term -4x = -12 Dividing both sides by -4 x=3 Substituting this new-found value for x into one of the original equations, the value of y is easily determined: x=3 substitute 2x + 2y = 14 6 + 2y = 14 Subtracting 6 from both sides 2y = 8 Dividing both sides by 2 y=4 Using this solution technique on a three-variable system is a bit more complex. As with substitution, you must use this technique to reduce the three-equation system of three vari- ables down to two equations with two variables, then apply it again to obtain a single equation with one unknown variable. To demonstrate, I’ll use the three-variable equation system from the substitution section:
  51. 51. 4.11. SOLVING SIMULTANEOUS EQUATIONS 43 x - y + z = 10 3x + y + 2z = 34 -5x + 2y - z = -14 Being that the top equation has coefficient values of 1 for each variable, it will be an easy equation to manipulate and use as a cancellation tool. For instance, if we wish to cancel the 3x term from the middle equation, all we need to do is take the top equation, multiply each of its terms by -3, then add it to the middle equation like this: x - y + z = 10 Multiply both sides by -3 -3(x - y + z) = -3(10) Distributive property -3x + 3y - 3z = -30 -3x + 3y - 3z = -30 (Adding) + 3x + y + 2z = 34 0x + 4y - z = 4 or 4y - z = 4 We can rid the bottom equation of its -5x term in the same manner: take the original top equation, multiply each of its terms by 5, then add that modified equation to the bottom equation, leaving a new equation with only y and z terms:
  52. 52. 44 CHAPTER 4. ALGEBRA REFERENCE x - y + z = 10 Multiply both sides by 5 5(x - y + z) = 5(10) Distributive property 5x - 5y + 5z = 50 5x - 5y + 5z = 50 (Adding) + -5x + 2y - z = -14 0x - 3y + 4z = 36 or -3y + 4z = 36 At this point, we have two equations with the same two unknown variables, y and z: 4y - z = 4 -3y + 4z = 36 By inspection, it should be evident that the -z term of the upper equation could be leveraged to cancel the 4z term in the lower equation if only we multiply each term of the upper equation by 4 and add the two equations together: 4y - z = 4 Multiply both sides by 4 4(4y - z) = 4(4) Distributive property 16y - 4z = 16 16y - 4z = 16 (Adding) + -3y + 4z = 36 13y + 0z = 52 or 13y = 52 Taking the new equation 13y = 52 and solving for y (by dividing both sides by 13), we get a value of 4 for y. Substituting this value of 4 for y in either of the two-variable equations
  53. 53. 4.12. CONTRIBUTORS 45 allows us to solve for z. Substituting both values of y and z into any one of the original, three- variable equations allows us to solve for x. The final result (I’ll spare you the algebraic steps, since you should be familiar with them by now!) is that x = 2, y = 4, and z = 12. 4.12 Contributors Contributors to this chapter are listed in chronological order of their contributions, from most recent to first. See Appendix 2 (Contributor List) for dates and contact information. Chirvasuta Constantin (April 2, 2003): Pointed out error in quadratic equation formula.
  54. 54. 46 CHAPTER 4. ALGEBRA REFERENCE
  55. 55. Chapter 5 TRIGONOMETRY REFERENCE Contents 5.1 Right triangle trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.1.1 Trigonometric identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.1.2 The Pythagorean theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2 Non-right triangle trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2.1 The Law of Sines (for any triangle) . . . . . . . . . . . . . . . . . . . . . . 48 5.2.2 The Law of Cosines (for any triangle) . . . . . . . . . . . . . . . . . . . . . 49 5.3 Trigonometric equivalencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.4 Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.5 Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.1 Right triangle trigonometry Hypotenuse (H) Opposite (O) Angle 90o x Adjacent (A) A right triangle is defined as having one angle precisely equal to 90o (a right angle). 47
  56. 56. 48 CHAPTER 5. TRIGONOMETRY REFERENCE 5.1.1 Trigonometric identities sin x = O cos x = A tan x = O sin x tan x = cos x H H A csc x = H sec x = H cot x = A cot x = cos x sin x O A O H is the Hypotenuse, always being opposite the right angle. Relative to angle x, O is the Opposite and A is the Adjacent. ”Arc” functions such as ”arcsin”, ”arccos”, and ”arctan” are the complements of normal trigonometric functions. These functions return an angle for a ratio input. For example, if the tangent of 45o is equal to 1, then the ”arctangent” (arctan) of 1 is 45o . ”Arc” functions are useful for finding angles in a right triangle if the side lengths are known. 5.1.2 The Pythagorean theorem H2 = A2 + O2 5.2 Non-right triangle trigonometry b C A a c B 5.2.1 The Law of Sines (for any triangle) sin a = sin b = sin c A B C
  57. 57. 5.3. TRIGONOMETRIC EQUIVALENCIES 49 5.2.2 The Law of Cosines (for any triangle) A2 = B2 + C2 - (2BC)(cos a) B2 = A2 + C2 - (2AC)(cos b) C2 = A2 + B2 - (2AB)(cos c) 5.3 Trigonometric equivalencies sin -x = -sin x cos -x = cos x tan -t = -tan t csc -t = -csc t sec -t = sec t cot -t = -cot t sin 2x = 2(sin x)(cos x) cos 2x = (cos2 x) - (sin2 x) 2(tan x) tan 2t = 1 - tan2 x sin2 x = 1 - cos 2x cos2 x = 1 + cos 2x 2 2 2 2 5.4 Hyperbolic functions ex - e-x sinh x = 2 ex + e-x cosh x = 2 sinh x tanh x = cosh x Note: all angles (x) must be expressed in units of radians for these hyperbolic functions. There are 2π radians in a circle (360o ). 5.5 Contributors Contributors to this chapter are listed in chronological order of their contributions, from most recent to first. See Appendix 2 (Contributor List) for dates and contact information.
  58. 58. 50 CHAPTER 5. TRIGONOMETRY REFERENCE Harvey Lew (??? 2003): Corrected typographical error: ”tangent” should have been ”cotan- gent”.
  59. 59. Chapter 6 CALCULUS REFERENCE Contents 6.1 Rules for limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.2 Derivative of a constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.3 Common derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.4 Derivatives of power functions of e . . . . . . . . . . . . . . . . . . . . . . . 52 6.5 Trigonometric derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.6 Rules for derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.6.1 Constant rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.6.2 Rule of sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.6.3 Rule of differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.6.4 Product rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.6.5 Quotient rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.6.6 Power rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.6.7 Functions of other functions . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.7 The antiderivative (Indefinite integral) . . . . . . . . . . . . . . . . . . . . . 55 6.8 Common antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.9 Antiderivatives of power functions of e . . . . . . . . . . . . . . . . . . . . . 56 6.10 Rules for antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.10.1 Constant rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.10.2 Rule of sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.10.3 Rule of differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.11 Definite integrals and the fundamental theorem of calculus . . . . . . . . 56 6.12 Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 51
  60. 60. 52 CHAPTER 6. CALCULUS REFERENCE 6.1 Rules for limits lim [f(x) + g(x)] = lim f(x) + lim g(x) x→a x→a x→a lim [f(x) - g(x)] = lim f(x) - lim g(x) x→a x→a x→a lim [f(x) g(x)] = [lim f(x)] [lim g(x)] x→a x→a x→a 6.2 Derivative of a constant If: f(x) = c Then: d f(x) = 0 dx (”c” being a constant) 6.3 Common derivatives d xn = nxn-1 dx d ln x = 1 dx x d ax = (ln a)(ax) dx 6.4 Derivatives of power functions of e If: If: x f(x) = e f(x) = eg(x) Then: Then: d f(x) = ex d f(x) = eg(x) d g(x) dx dx dx

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